Constant mean curvature one surfaces in hyperbolic 3-space using the Bianchi-Calò method

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Constant mean curvature one surfaces in hyperbolic

3

-space

using the Bianchi-Calò method

LEVI L. DE LIMA and PEDRO ROITMAN

Departamento de Matemática, Universidade Federal do Ceará Campus do Pici, 60455-760 – Fortaleza, Brasil

Manuscript received on October 29, 2001; accepted for publication on November 27, 2001; presented byManfredo do Carmo

ABSTRACT

In this note we present a method for constructing constant mean curvature on surfaces in hyper-bolic 3-space in terms of holomorphic data first introduced in Bianchi’s Lezioni di Geometria Differenziale of 1927, therefore predating by many years the modern approaches due to Bryant, Small and others. Besides its obvious historical interest, this note aims to complement Bianchi’s analysis by deriving explicit formulae for CMC-1 surfaces and comparing the various approaches encountered in the literature.

Key words: Constant mean curvature one surfaces, congruence of spheres, rolling of surfaces, Weierstrass representation.

1 INTRODUCTION

It is generally accepted that the theory of surfaces in hyperbolic 3-space

H

3

₍

−

1

)

with constant

mean curvature equal to one (CMC-1 surfaces, for short) started with a seminal paper by R. Bryant

(Bryant 1987), where he derives a representation for such surfaces in terms of holomorphic data

in analogy with the well-known Weierstrass representation for minimal surfaces in

R

3

_(Lawson

1980).

After the appearance of Bryant’s investigation, many other researchers contributed to the

sub-ject. For example, M. Umehara and K. Yamada (Umehara and Yamada 1993) refined substantially

Bryant’s approach and were able to construct a varied class of examples of CMC-1 surfaces, besides

developing many interesting global aspects in the theory. On the other hand, A. J. Small (Small

Correspondence to: Levi L. de Lima

1994) reinterpreted CMC-1 surfaces in terms of twistors, providing in particular a Weierstrass

for-mula for such surfaces involving algebraic operations on the derivatives up to second order of a

pair

(f, g)

of holomorphic functions (see (8) below).

Consensual as it may be, the above account is not entirely correct and the main purpose of

this note is to stress that in Bianchi’s Lezioni di Geometria Differenziale (Bianchi 1927), we may

find a recipe for constructing CMC-1 surfaces from holomorphic data. However, the motivation

for writing this note goes beyond this historical curiosity, for it seems that there are at least two

other reasons for exhibiting this old method to a wider audience.

The first one is that the method allows one to start with an arbitrary holomorphic map

f

=

f (z)

defined in a region

⊂

C

_{and, elaborating upon Bianchi’s ideas, to end up with explicit formulae}

for a CMC-1 surface. More precisely, if we use the upper half-plane model for

H

3

₍

−

1

)

with

Cartesian coordinates

(x

1

, x

2

, x

3

)

so that the boundary at infinity is given by

x

3

=

0, we have

Theorem

1

_.

1

.

_{In the above situation, the parametrization of a CMC-1 surfaces in terms of}

_f

_is

given by

x

1

=

Re f

−

f

′

2

Re

f

′

z

+

1+|₂z|2

Re

(f

′

)

2

f

¯

′′

|

f

′

_|

2

₊

_Re

f

′

_f

¯

′′

_z

_¯

+

|f′′|2(|₄z|2+1)

,

x

2

=

I mf

−

f

′

2

I m(f

′

z)

+

1+|₂z|2

I m

(f

′

)

2

f

¯

′′

|

f

′

_|

2

₊

_Re

f

′

_f

¯

′′

_z

_¯

+

|f′′|2(|₄z|2+1)

,

(1)

x

3

=

f

′

3

|

f

′

_|

2

₊

_Re

f

′

_f

¯

′′

_z

_¯

+

|f′′|2(|₄z|2+1)

.

Moreover, this map

f

has an immediate geometric interpretation: it is simply the parametrized

hyperbolic Gauss map, or in other words, the expression for the hyperbolic Gauss map in terms

of a local complex parameter

z

on

. In fact, and this is an important issue here, our formulae

coincide with Small’s if the pertinent transformation between models for

H

3

₍

−

1

)

is carried out

(see Section 3).

The second reason is that in his way toward the construction of CMC-1 surfaces, Bianchi

translates to hyperbolic geometry the solution of a strictly Euclidean-geometric problem involving

the rolling of a pair of isometric surfaces, thereby establishing a surprising linking between these

two geometries.

2 CONGRUENCE OF SPHERES AND ROLLING OF SURFACES

Here is the first classical concept we shall meet. A

congruence of spheres

is a smooth two-parameter

family of spheres in

R

3

_,

that we will suppose parametrized by coordinates

_{(u, v)}

. To each such

congruence we may associate a function

R

=

R(u, v)

, the

radius function

, describing the radii of

the spheres in the congruence. We also assume that the vector function

X

=

X

(u, v)

describing

the centers of the spheres defines a regular surface which we call the

surface of centers

.

Generically there are two surfaces, the so-called

envelopes

, associated to a given congruence.

In effect, a point

p

∈

R

3

_{belongs to an envelope}

_ξ

_if

_p

∈

S

for some sphere

S

in the congruence

and moreover

T

p

ξ

=

T

p

S

. If a congruence of spheres has two distinct envelopes we then have

a natural correspondence between their points, namely, points on distinct envelopes correspond

if they are the contact points of the envelopes with a given sphere of the congruence. The next

proposition gives the expression for the envelopes in terms of the unit normal vector

N

=

N

(u, v)

and the metric of the surface of centers.

Proposition

2

_.

1

_.

_{In coordinates}

_{(u, v),}

ξ

=

X

−

R

(

X

, R)

±

1

−

1

R

N

,

(2)

where

(

X

, R)

=

(R

u

A

11

+

R

v

A

12

)

X

u

+

(R

u

A

21

+

R

v

A

22

)

X

v

,

(3)

and

1

R

=

R

2u

A

11

+

2

R

u

R

v

A

12

+

R

2v

A

22

.

(4)

Here, the matrix

A

= [

A

ij

]

is the inverse of the matrix defined by the metric in the given coordinates.

We now describe the rolling of isometric surfaces. Consider a pair

(S,

S)

˜

of isometric surfaces

in

R

3

_,

and let

_p

∈

S

and

p

˜

∈ ˜

S

be points corresponding under the isometry. Suppose

S

is fixed

in space and consider the two-parameter family of positions of congruent copies of

_S

˜

_{such that}

to each

p

∈

S

we consider a rigid motion of

R

3

(call it

_H

_p

) sending

p

˜

to

p

,

T

p˜

S

˜

to

T

p

S

, and

further adjusted so that the differential of the isometry composed with

H

p

is the identity map. This

two-parameter family of positions for copies of

_S

˜

_{is called the}

_rolling

_of

_S

˜

_over

_S

_{. The surface}

_S

˜

is called the

rolled surface

and

S

the

support surface

.

Now fix a point

O

∈

R

3

and consider its image under the two-parameter family of rigid motions

associated to the rolling of

_S

˜

_over

_S

_{. In the generic case, the motion of}

_O

_{defines a surface}

_called

the

rolling surface

with respect to the

satellite point

O

.

The crucial point now is that the two concepts introduced so far, namely congruence of spheres

and rolling of surfaces, share a close relationship. More precisely, we have

Proposition

₂

_.

₂

.

_{Given a rolling of}

_S

˜

_over

_S

_{as above, the rolling surface}

_{can be viewed}

We are now in a position to formulate the problem in Euclidean geometry whose solution will

lead us, according to Bianchi, to a method for constructing CMC-1 surfaces in

H

3

₍

−

1

)

:

Find pairs

(S,

S)

˜

of isometric surfaces such that, for a convenient satellite point

O

,

the rolling

surface

is contained in a plane.

Surprisingly enough, assuming that the plane in question is

{

x

3

=

0

}

and moreover that the

satellite point

O

is the origin of our coordinate system, this problem admits a neat solution in terms

of an arbitrary holomorphic function

f

=

f (z)

. More precisely, the radius function

R

is given by

R

=

1

+ |

z

|

2

2

f

′

(z)

,

(5)

and the solution to our problem, namely, the coordinates of

_S

˜

_and

_S

_{are respectively given in terms}

of the holomorphic data as

˜

x

=

f

′

(z)

z

+ ¯

z

2

,

y

˜

=

f

′

(z)

z

− ¯

z

2

i

,

z

˜

=

f

′

(z)

|

z

|

2

−

1

2

,

(6)

and

x

=

Re

f (z),

y

=

Im

f (z),

z

=

f

′

(z)

|

z

|

2

+

1

2

.

(7)

The above expressions are called

Calò’s formulae

since they have been originally published

by B. Calò in 1899 (Calò 1899) in another context involving isometric surfaces.

3 THE BIANCHI-CALÒ METHOD

In the last section, starting with a holomorphic map

f

, we have determined a pair of isometric

surfaces such that one of the envelopes of the associated congruence of spheres was a plane. Now, in

principle we could also determine the second envelope of the congruence associated to the rolling,

which is then also contained in the upper half-space. It can be shown that the correspondence

between the envelopes of the congruence associated to the Calò’s pair

(S,

S)

˜

considered in the

last section is a conformal map. This is proved in Bianchi’s Lezioni when he considers Darboux

congruencies and is one of the ingredients in the proof of the following central result, also due to

Bianchi.

Theorem

3

_.

1

_.

_{To each pair}

₍

_{S, S)}

˜

_{of isometric surfaces such that the rolling surface}

_{of the}

rolling of

S

˜

over

S

is a plane there corresponds a CMC-1 given by the second envelope of the

associated congruence of spheres considered as a surface in the standard upper half-space model

of

H

3

₍

₋

₁

_).

map

G

, which is precisely the correspondence between the envelopes. On the other hand,

G

is

known to be conformal, exception made for totally umbilical surfaces, exactly when the surface is

a CMC-1 surface (see (Bryant 1987) or (Bianchi 1927)), and this concludes the proof.

Although Bianchi indicates how one can find CMC-1 surfaces starting with an arbitrary

holo-morphic map

f

via Theorem 3.1, he does not complete his analysis by deriving explicit formulae.

However, this can be carried out very simply: we use Calò’s formulae (5) and (7) to compute the

surface of centers

S

and the radius function

R

, then we calculate, by means of (2), the envelopes of

this congruence of spheres, one of them being a piece of the plane

{

z

=

0

}

and the other one being

our CMC-1 surface given by (1). This ends the sketch of the proof of Theorem 1.1.

Remark

3

_.

2

_.

It is easy to check that

_f

=

G

◦

X

, where

X

=

(x, y, z)

is given by (1) and

G

is the

hyperbolic Gauss map

G

of the corresponding CMC-1 surface.

Now we briefly indicate how Small’s result, obtained by using a heavy algebraic-geometric

machinery, relates to the one presented here. Small works in the so-called hermitian model for

H

3

₍

−

1

)

and introduces, via twistor theory, a Gauss transform

Ŵ

S

=

(f, g)

of a null curve

S

⊂

SL(

2

,

C

₎

_{, where}

_SL(

₂

_,

C

₎

_{is the orientation preserving isometry group of}

H

3

₍

−

1

)

, in terms of

a pair

(f, g)

of holomorphic functions. His main result is that

S

can be recovered from

_Ŵ

S

after

applying dualization. After doing this, one realizes that

S

is given by the map

_ω

:

M

→

SL(

2

,

C

₎

_,

ω

=

α

β

γ

δ

=

(f

′

₎

1/2

₋

1

2

f (f

′

)

−3/2

f

′′

f

(f

′

₎

−1/2

₊

1

2

g(f

′

)

−3/2

f

′′

−

g(f

′

₎

1/2

−

1₂

(f

′

)

−3/2

_f

′′

_(f

′

₎

−1/2

₊

1

2

g(f

′

)

−3/2

f

′′

,

(8)

where

f

′

₌

_df/dg

_and

_f

′′

₌

_d

2

_f/dg

2

_{. The expression for the CMC-1 surface in terms of the}

hermitian model is given by

ωω

t

, but in order to compare this with (1) one has to perform the

transformation to the upper half-space model. In terms of the entries of

ω

, this is given by

x

1

+

ix

2

=

αγ

+

βδ

|

γ

|

2

+ |

δ

|

2

, x

3

=

1

|

γ

|

2

+ |

δ

|

2

.

We now take

g

=

z

and Small’s

f

to be our

f

. A straightforward computation yields the equivalence

between the methods.

We illustrate the method by retrieving two well-known examples. First, if we take

f (z)

=

z

2

,

substitution in (1), after writing

z

=

re

iθ

, yields

X

=

−

r

2

(

cos 2

θ )

5

r

2

₊

₃

7

r

2

+

1

,

−

r

2

₍

_{sin 2}

_{θ )}

5

r

2

+

3

7

r

2

+

1

,

8

r

3

7

r

2

+

1

.

This is a

catenoid cousin

.

Now let

f (z)

=

ln

z.

Again, substitution in (1) yields

X

=

ln

r

−

2

(r

−

r

−1

₎

(r

+

r

−1

₎

, θ,

4

(r

+

r

−1

₎

Or, writing

r

=

e

s

,

X

=

s

−

2 tanh

s, θ,

2

cosh

s

.

This is a ruled example.

RESUMO

Nesta nota apresentaremos um método para construir superfícies de curvatura média constante um no 3-espaço hiperbólico, a partir de funções holomorfas. Tal método foi introduzido nas Lezioni di Geometria Differenziale de Bianchi em 1927, antecedendo, portanto, em muitos anos, os pontos de vista mais modernos de Bryant, Small e outros. Além do seu óbvio interesse histórico, o objetivo da nota é complementar a análise de Bianchi, obtendo fórmulas explícitas para as superfícies de curvatura média constante um, e comparar os vários pontos de vista encontrados na literatura.

Palavras-chave:Superfícies de curvatura média constante um, congruência de esferas, rolamento de

super-fícies, representação de Weierstrass.

REFERENCES

Bianchi L.1927.Lezioni di Geometria Differenziale, Terza Edizione, Nicola Zanichelli Editore, Bologna.

Bryant R. _{1987. Surfaces with constant mean curvature one in hyperbolic space, Astérisque 154-55:} 321-347.

Calò B.1899. Risoluzione di alcuni problemi sull’applicabilità delle superficie, Annali di Matematica IV.

Lawson HB._{1980. Lectures on minimal submanifolds I. Mathematics Lecture Series 9, Publish or Perish.}

Small AJ.1994. Surfaces of constant mean curvature one inH3and algebraic curves on a quadric, Proc. of the A.M.S. 122: 1211-1220.